Asymptotic behavior for a class of delay differential equations with a forcing term

Abstract

We study the asymptotic behavior of solutions of the following forced delay differential equation 

$$ x'(t)=-p(t)f(x(t-\tau))+r(t),\quad t\ge 0.  \eqno{(*)}$$ 

It is show that if $ f$ is increasing and $ |f(x)|\le |x|$ for all $ x\in R$, $ \lim_{t\to +\infty} {r(t)\over p(t)}=0$, $ \int_0^{+\infty} p(s)ds=+\infty$ and $ \limsup_{t\to+\infty} \int_{t-\tau}^t p(s)ds <{3\over 2}$ for sufficiently large $t$, then every solution of the Eq.$(*)$ tends to zero as $t$ tends to infinity. Our result improves the recent results obtained by Graef and Qian.